We inspected the MXDF data for nine sources that are not in the MXDF catalog. Seven sources of those nine sources at \(z_{\rm p}=0.06^{+0.04}_{-0.05}\) , \(0.82^{+0.07}_{-0.55}\) , \(1.11^{+0.14}_{-0.58}\) , \(1.63^{+0.12}_{-0.13}\) , \(2.03^{+0.15}_{-0.16}\) , \(2.43^{+0.13}_{-0.54}\) , and \(2.55^{+0.16}_{-0.17}\)  show clear continua at the blue edge of the MUSE spectra, which is not the case at \(z\ge 2.86\) . One of the remaining two sources, RID=54891, has \(z_{\rm p}=2.31^{+0.18}_{-0.19}\)  and does not have an ORIGIN-detected emission line. However, it shows weak potential \(\mbox{Ly$\alpha $}\)  emission at \(z=2.94\)  with a line flux S/N of \(3.9\) . Because of the low S/N, the noisy spectrum, and the missmatch between \(z_{\rm s}\)  and \(z_{\rm p}\) , it was not categorized as a ZCONF=1 source and was not included in the MXDF catalog. To minimize a sample selection bias in this work, we assigned it a tentative spec-\(z\)  of \(z_{\rm s}=2.94\) . This object is not included in the UV-bright sample described later in this Section. We confirmed that our main results can hold up with the subset. The other remaining source at \(z>2.86\)  that is not in the MXDF catalog (RID=6693) has \(z_{\rm p}=4.21^{+0.25}_{-0.27}\)  and shows a break around 6250 Å in the MUSE spectrum, which can be interpreted as a Ly\(\alpha \)  break. The source has an emission line in the 1D spectrum, which could be interpreted as a \(\mbox{Ly$\alpha $}\)  emission line at \(z\simeq 4.17\) , but is contamination from a neighboring LAE. Then, we performed a cross correlation using the MARZ spec-\(z\)  software [1]}, [2]} on different spectra extracted from 1 to 5-pixel radius apertures and did not find realistic solutions. In order to obtain our best estimate of a tentative spec-\(z\) , we stacked the 1D spectra centered at wavelengths of different combinations of UV absorption lines among Si ii\(\lambda \) 1260.42, O i\(\lambda \) 1302.17,Si ii\(\lambda \) 1304.37, C ii\(\lambda \) 1334.53, Si iv\(\lambda \lambda \) 1393.75,1402.77, Si ii\(\lambda \) 1526.71, C iv\(\lambda \lambda \) 1548.20,1550.78, Fe ii\(\lambda \) 1608.45, and Al iii\(\lambda \) 1670.79. It is challenging to determine a spec-\(z\)  for faint sources at high redshifts with this method, but we note that we can get correct spec-\(z\)  for brighter sources at \(z>4\)  with ZCONF=3, RID\(=5479\)  and 7067. Although we did not find a significant absorption line at any redshift solution, \(z_{\rm s}=4.167\)  shows 3 consecutive pixels with more than a \(1\sigma \)  dip compared to the continuum (\(1.2\sigma \) , \(1.8\sigma \) , and \(1.6\sigma \) ) as a result of stacking of Si ii\(\lambda \) 1260.42, C ii\(\lambda \) 1334.53, and Si iv\(\lambda \) 1393.75 lines. This is consistent with the interpretation of the Ly\(\alpha \)  break. We note that Si iv\(\lambda \) 1402.77 at \(z_{\rm s}=4.167\)  overlaps with a skyline. Therefore, we assigned a tentative spec-\(z\) , \(z_{\rm s}=4.167\) . We did not discard these two objects (RID=54891 and 6693) to minimize a selection bias. RID=6693 can meet criteria for a UV-bright subsample described later in this Section, while RID=54891 does not, which means that it does not affect our main conclusion. We took into account the effect of uncertainties in the spec-\(z\)  estimation (see Section REF ).
